By looking at Fig. 1, which compares the instantaneous values of the second invariant of the velocity gradient tensor of the laminar and turbulent inlet simulations, we note that the coherent structures generated by Kelvin-Helmholtz instabilities are only observable in the free jet region of the laminar inlet case. On the other hand, the flow in the free jet shear layer appears strongly more chaotic in the turbulent inlet case. Although the wall jet region appears fully turbulent in both cases, we observe a recurrent aggregations of vortices flowing downstream (vortex rings) only in the laminar inlet case. As shown in Fig. 2, this reflects in the mean Nusselt number distribution at the wall, where the characteristic shoulder is not observable in the turbulent inlet case. We found that in the latter case, the Nusselt number is approximately 20% higher in the region r/D<1.2, whereas the laminar inlet case features a 5% larger heat flux in r/D>1.8. The surprising result is that both jets provide about the same heat flow rate (in the region r/D<4), because the region where the turbulent inlet jet has a much better cooling rate (approx. 20% higher) is much smaller, too (approx. 2.5 times smaller).

Fig. 2: Average Nusselt number distribution at the wall as a function of the dimensionless distance from the jet axis r/D (turbulent inlet case).

Root mean squares (RMS) of the velocity in the shear layer are in good agreement when experimental and numerical data are compared. In the laminar inlet case, as a result of the laminar shear layer profile, natural modes of the impinging jet can develop undisturbed. In particular, modal structures appear in form of axis-symmetric ring vortices, which are accompanied with strong RMS of axial and radial velocity fluctuations in the free and wall jet regions. In the experiments, however, several different modes are induced by the turbulent inlet. They prevent the free development of natural modes so that the ring vortices in the experiment occur only in weakened and irregular form. As a result, the corresponding RMS in the shear layer areas are smaller.

In the 1st funding period we carried out a dynamic mode decomposition (DMD) of our DNS flow field. With this method, a dominant mode which has same non-dimensional frequency as the most effective inlet pulsation in the experiment could be detected. It is known that the application of the DMD in a linear system gives the eigenfunctions of the system. The linear stability analysis of the impinging jet system showed that global modes exist, whose the frequencies of correspond to the DMD frequencies. It is noteworthy that the sensitivity to external forcing is attained in the proximity of the jet inlet (Fig. 3). This feature can be gainfully exploited for the purpose of devising cooling efficiency enhancement methods.

Fig. 4: Average Nusselt number Nu on the target plate as a function of the non-dimensional distance from jet axis r/D (curved plate case).

In order to resemble the internal configuration of a gas turbine blade, a DNS of a jet impinging on flat plate has been performed. The analysis focused on assessing how good is the common hypothesis by which curved plates can be approximated as flat when estimating the heat transfer efficiency through them. It has been found that integral heat flux (integrated on the plate up to a distance from the jet axis of 5 jet diameters) differs from the flat plate case (reference case) by just 0.03% (Fig. 4). On the other hand, peak frequencies of the instantaneous Nusselt number appear reduced by 40% and 50% when compared to the flat plate case. A dynamic mode decomposition (DMD) of the system showed, as observed in the reference case, that the modes oscillating at the aforementioned peak frequencies are those responsible for the characteristic heat flux profile at the wall. It can be consequently affirmed that if only averaged properties are of interest, it is possible to model curved impingement plates as flat surfaces. On the contrary, when the dynamic response of the system is addressed in order to implement, for instance, dynamic heat transfer enhancement techniques, the presence of the curved plate can not be disregarded.

Analysis of a linear array of impinging jets in cross flow

5. Shape of the most dominant mode for the cross-flow case, depicted with an isosurface of Q.

DNS of an infinite linear array of impinging jets is carried out with the addition of cross flow. This configuration mimics a line of jets in the spanwise direction of the blade which follows the first line and is therefore subject to a cross flow induced by the previous lines of jets. Reynolds and Mach numbers of the simulation are respectively 8000 and 0.8, whereas the nozzle-to-plate distance is, as in the previous case, 5D. Similarly to the experiments carried out in B03, the blowing ratio is set to 5. Also in this case, the DMD of the flow indicates that two dominant modes exist. By looking at Fig. 5 we observe that the first and second mode are not perfectly toroidal because of the effect of the cross flow. Nevertheless, the cross flow does not prevent the formation of Kelvin-Helmholtz instabilities, which are particularly correlated with the first dominant mode. As in the absence of cross flow, we expect that pulsating the jet with a frequency close to that of the dominant mode is optimal. Importantly, the latter reduces by 15% when compared to the case without cross flow. The frequency associated to the second mode is affected as well, reducing in the same proportion. Analogously to the curved plate case, we conclude that the cross flow plays a role that is not negligible for the correct design of dynamic internal cooling techniques for gas turbine applications.

Acknowledgement

Simulations were performed at the High Performance Computing Center Stuttgart (HLRS) under the grant number JetCool/44127.